Search Results (207)

The Hilfer fractional derivative effectively captures non-locality, historical dependence, and memory effects, making it valuable for modeling real-world systems, and exponential growth can describe explosive growth phenomena in real-world problems. This paper focuses on the existence of mild solutions for infinite-delay differential equations involving Hilfer fractional derivatives, fractional Laplacian operator ${(-\Delta )}^{\delta }$, and exponentially growing functions in Orlicz spaces. First, by utilizing standard $L^p$-$L^q$ estimates for strongly continuous semigroups generated by fractional Laplacian operator, the existence of global solutions in the Orlicz space $\exp L^p\left({\mathbb{R}}^d\right)$ and the time-weighted $L^z\left({\mathbb{R}}^d\right)$ space is established. Then, by leveraging Hölder’s interpolation inequality, the existence of local solutions in $L^1\left({\mathbb{R}}^d\right)\cap L^{\infty }\left({\mathbb{R}}^d\right)$ is established. Full article

In this paper, based on the Schauder fixed point theorem, the (generalized) Darbo fixed point theorem, and the (generalized) Banach contraction mapping principle, we study the mixed-type fractional impulse evolution equation with non-local and delay terms, and obtain the existence and uniqueness theorems under whether the operator is compact or not. The order of the derivative in this paper is $0<\alpha <1$, this fractional order introduces a series of problems concerning compactness, continuity, and convergence. We overcome these problems using methods such as Hölder inequality and Minkowski inequality. Moreover, under the condition of the non-compact measure, the non-negative constant is extended to an unbounded Lebesgue-integrable function. In addition, when obtaining the uniqueness of the solution through the (generalized) Banach contraction mapping principle, the non-negative constant L in the Lipschitz condition is extended to an unbounded Lebesgue integrable function. Finally, a case study is conducted to demonstrate the validity of the theoretical results. Full article

Background/Objectives: Women+ (e.g., women and individuals assigned female at birth) experience disproportionate health risks and persistent gaps in access to care. Women+ health research remains significantly underfunded and understudied, contributing to inequities in diagnosis, treatment, and outcomes. This study aims to collaboratively identify and prioritize the most pressing unanswered research questions related to women+ health in the Maritime provinces of Canada. Methods: This study will use a modified Priority Setting Partnership (PSP) methodology based on the James Lind Alliance framework. A mixed-methods participatory approach will be used, including bilingual online surveys (French, English) and a one-day consensus workshop. Participants will include women+, healthcare professionals, researchers, policymakers, and the public residing in the Maritime provinces (Nova Scotia, New Brunswick, and Prince Edward Island). An initial survey will collect research uncertainties through open-ended questions. A second survey will rank verified uncertainties, followed by a facilitated workshop to achieve consensus on the Top 10 research priorities. Descriptive statistics will summarize participant demographics. Anticipated Results: This project is expected to generate a collaboratively developed Top 10 list of research priorities for women+ health in the Maritimes, which will be used to prioritize future research related to women+ health. Conclusions: By centering women+ voices and engaging diverse interest holders, this study will establish a shared regional research agenda to guide future research, funding, and policy initiatives for women+ health research. Full article

In this paper, we present a novel identity for twice partially differentiable mappings. Based on this identity, new fractional Bullen-type inequalities for differentiable functions of two variables, which are convex on the coordinate via Riemann–Liouville fractional integral operators are derived. Other results are obtained by applying integral inequalities, including the Hölder, the improved Hölder, and the power mean inequalities. We apply these findings to special means. A numerical example with graphical illustrations is presented to demonstrate the validity and effectiveness of our theoretical findings. Full article

The goal of this study is to establish a new type of Milne-type inequality in the scope of fractional calculus with the aid of proportional Caputo-hybrid operators. We will focus on two different scopes of regularity, which contain functions whose first and second derivatives are convex, and functions whose first and second derivatives are Lipschitz continuous. We will base these estimates on a new integral identity of proportional Caputo-hybrid integrals. We will show that the smoothness of the derivative influences the shape of the bounds. Convexity will cause symmetry. Lipschitz continuity will contain bounds on the modulus of continuity. To show that our results are accurate and easy to obtain, we included a full numerical example with graphics and applications to quadrature error estimation. Full article

Accuracy in fractional numerical integration is often limited by the regularity of the integrand. This work proposes a flexible error estimation framework for proportional Caputo-hybrid integral operators based on $\mathfrak{s}$-convexity. We introduce a parametric Newton–Cotes formula ($\nu \in [0,1]$) that bridges the gap between classical quadrature rules, recovering the fractional Trapezoidal, Midpoint, and Simpson’s methods as specific instances. In order to confirm the correctness of our results, we provide an illustrative example with graphical representations. Furthermore, we provide some additional results using Hölder’s and power mean inequalities and employ a verification strategy based on an Artificial Neural Networks (ANNs) model. The ANN approach allows for high-dimensional parameter space exploration, demonstrating that the proposed inequalities provide robust and precise error estimates. Full article

This paper introduces a comprehensive class of multiparameter post-quantum fractional quadrature inequalities, unifying classical error bounds within the setting of the post-quantum Riemann–Liouville fractional integral. By incorporating multiple parameters, we derive a flexible family of inequalities that generalize well-known quadrature rules such as the Boole-type, Bullen–Simpson-type, Maclaurin-type, corrected Euler–Maclaurin-type, ${\displaystyle \frac{3}{8}}$-Simpson-type, and companion Ostrowski-type estimates. Under assumptions of s-convexity, log-convexity, power mean inequality, and Holder inequality, we establish novel error bounds. Our results provide a unified framework for designing and analyzing post-quantum fractional quadrature inequalities. Applications to special means and numerical and graphic examples are presented to illustrate the applicability and generality of the derived inequalities. This work lays a theoretical foundation for the development of post-quantum fractional quadrature inequalities and offers new tools for error estimation in post-quantum fractional-order models arising in applied sciences and engineering. Full article

The Minkowski and Hölder inequalities play an important role in many areas of pure and applied mathematics, such as Convex Analysis, Probabilities, Control Theory, Fixed Point theorems, and Mathematical Economics. Also, non-additive measures, non-additive integrals and set-valued integrals are useful tools in several areas of theoretical and applied mathematics. In this paper we present and prove some Hölder and Minkowski inequality (or reverse inequality) types obtained for Birkhoff weak integrable functions with respect to a non-additive measure. Then, we apply these results to the interval-valued case. Full article

Background: Immigrant and racialized families raising children on the autism spectrum in Canada navigate intersecting inequities shaped by racism, language barriers, immigration status, and fragmented health and education systems. Yet their perspectives remain underrepresented in autism and health policy research. Methods: Guided by the socioecological and critical social science lens, this community-based participatory study employed a PhotoVoice approach in partnership with SMILE Canada–Support Services. Ten immigrant and/or racialized family caregivers from the Greater Toronto area participated in four in-person sessions involving ethical training, guided photo-taking, group-based reflections, and collaborative theme refinement. The data included 38 participant-generated photographs, narratives, and an audio-recorded final group discussion. Results: Seven interrelated themes were identified: (1) family support and child needs; (2) physical and emotional burden on caregivers; (3) school support or its missingness; (4) stigma and discrimination; (5) overall journey with barriers; (6) transitions and uncertainty; and (7) two sides of a coin: isolation and strength, loneliness and hope. Caregivers highlighted extensive invisible labor, exclusionary schooling, financial and systemic barriers, and cumulative stress. Simultaneously, they articulated resilience, mutual support, and a strong sense of collective responsibility. The PhotoVoice process itself was experienced as validating, unifying, and empowering, with participants expressing readiness to disseminate findings through exhibitions, school boards, universities, and policy-engagement initiatives. Conclusions: Caregiving among immigrant and racialized families is both a profound act of love and a site of structural injustice. Centering on caregivers as co-researchers and knowledge holders reveals urgent needs for equity-oriented autism policies and culturally responsive, accessible support systems in Canada. Full article

This study investigates the relationship between stock returns and income inequality in South Africa, a country marked by persistently high levels of income disparities and a sophisticated and structurally unique financial market. Despite the Johannesburg Stock Exchange (JSE) being one of the most developed and liquid markets in Africa, stock ownership remains limited to a small segment of the population, often reinforcing pre-existing income inequalities. This study determines the relationship between stock returns and income distribution using the ARDL bound test methodology. Using time series data from 1975 to 2024, the study examines the extent to which stock market returns influence income distribution. The findings of the study suggest a positive relationship between stock returns and income distribution. This relationship suggests that higher stock market development disproportionately benefits capital holders. The long-term relationship seems to have limited feedback from inequality to stock returns. The findings aim to inform policies on inclusive financial participation and broad-based wealth generation to address South Africa’s structural inequalities. Full article

The purpose of this research is to develop a set of Hermite–Hadamard–Mercer-type inequalities that involve different types of fractional integral operators such as classical Riemann–Liouville fractional integral operators. Furthermore, some fractional integral inequalities are obtained for three-times differentiable convex functions with respect to the right-hand side of the Hermite–Hadamard–Mercer-type inequality. Moreover, several new results regarding Young’s inequality, bounded function and L-Lipschitzian function are deduced. The paper presents additional remarks and comments on the results to make sense of them. To illustrate the key findings, graphical representations are provided, and applications involving special means, midpoint formula, q-digamma function and modified Bessel function are presented to demonstrate the practical utility of the derived inequalities. Full article

The goal of this paper is to use a Boole-type inequality framework to provide better estimates for differentiable functions. Using majorization theory, fractional integral operators are incorporated into a new auxiliary identity. The method establishes sharp bounds by combining the properties of convex functions with classical inequalities like the Power mean and Hölder inequalities, as well as the Niezgoda–Jensen–Mercer (NJM) inequality for majorized tuples. Additionally, the study presents real-world examples involving special functions and examines pertinent quadrature rules. This work’s primary contribution is the extension and generalization of a number of results that are already known in the current body of mathematical literature. Full article

This work presents refined and generalized forms of weighted Opial-type inequalities within the framework of time scale calculus. The proofs rely on several algebraic techniques, together with Hölder’s inequality and Keller’s chain rule. These results extend the classical Opial-type inequalities by embedding them into the time scale setting, which unifies both continuous and discrete analyses. Consequently, various integral and discrete inequalities emerge as particular cases of our main results, thereby broadening the applicability of Opial-type inequalities to dynamic systems and discrete models. Full article

In this paper, we establish some Simpson-type inequalities within the framework of Riemann–Liouville fractional calculus, specifically tailored for differentiable harmonically convex functions. By introducing a novel fractional integral identity for differentiable functions with harmonic arguments, we derive several estimates that generalize and refine existing results in the literature. The theoretical findings are validated through a numerical example supported by graphical illustration, and potential applications in approximation theory and numerical analysis are discussed. Full article

By leveraging the concept of k-Caputo fractional derivatives for stochastic processes, in this paper, we derive a generalized Hermite–Hadamard inequality tailored to n-times differentiable convex stochastic processes, providing a powerful tool for analyzing systems governed by fractional dynamics in probabilistic settings. Additionally, we establish two new integral identities that serve as the foundation for developing midpoint- and trapezium-type inequalities for $(n+1)$-times differentiable convex stochastic processes. These results not only enrich the theoretical underpinnings of fractional calculus, but also offer practical implications for modeling and understanding complex systems with memory and randomness. The proposed framework opens new avenues for future research in stochastic analysis and fractional calculus, with potential applications in fields such as financial mathematics, engineering, and physics. Full article